Simplify the following expression and state the condition under which the simplification is valid: $a = \dfrac{z^2 - 5z - 6}{z^2 - 6z}$
Solution: First factor the expressions in the numerator and denominator. $ \dfrac{z^2 - 5z - 6}{z^2 - 6z} = \dfrac{(z + 1)(z - 6)}{(z)(z - 6)} $ Notice that the term $(z - 6)$ appears in both the numerator and denominator. Dividing both the numerator and denominator by $(z - 6)$ gives: $a = \dfrac{z + 1}{z}$ Since we divided by $(z - 6)$, $z \neq 6$. $a = \dfrac{z + 1}{z}; \space z \neq 6$